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Approximately bisectrix-orthogonality preserving mappings

100%

Zamani A.

Commentationes Mathematicae

2014

tom 54

nr 2

Regarding the geometry of a real normed space ${\mathcal X}$, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors $x, y \in {\mathcal X}$ as follows:$${x\sideset{ ^{\varepsilon\!\!}}{}\perp}_W y \mbox{~if and only if~} \sqrt{2}\frac{1-\varepsilon}{1+\varepsilon}\|x\|\,\|y\|\leq \Big\|\,\|y\|x+\|x\|y\,\Big\|\leq\sqrt{2}\frac{1+\varepsilon}{1-\varepsilon}\|x\|\,\|y\|.$$ We study the class of linear mappings preserving the approximately bisectrix-orthogonality ${\sideset{ ^{\varepsilon\!\!}}{}\perp}_W$. In particular, we show that if $T: {\mathcal X}\to {\mathcal Y}$ is an approximate linear similarity, then $${x\sideset{ ^{\delta\!\!}}{}\perp}_W y\Longrightarrow {Tx \sideset{ ^{\theta\!\!}}{}\perp}_W Ty \qquad (x, y\in {\mathcal X})$$ for any $\delta\in[0, 1)$ and certain $\theta\geq 0$.

Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

88%

Stypuła T., Wójcik P.

Annales Mathematicae Silesianae

2016

tom 30

nr 1

193-201

In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

Ograniczanie wyników

1 Annales Mathematicae Silesianae

1 Commentationes Mathematicae

1 Stypuła T.

1 Wójcik P.

1 Zamani A.

1 2016

1 2014

Wyniki wyszukiwania

Approximately bisectrix-orthogonality preserving mappings

Characterizations of Rotundity and Smoothness by Approximate Orthogonalities